The Generalized Dynamic-Factor Model: Identification …pages.stern.nyu.edu/~dbackus/Identification/FHLR_factor_REStat_00.pdf · The Generalized Dynamic-Factor Model: Identification

The Generalized Dynamic-Factor Model: Identification and EstimationAuthor(s): Mario Forni, Marc Hallin, Marco Lippi and Lucrezia ReichlinSource: The Review of Economics and Statistics, Vol. 82, No. 4 (Nov., 2000), pp. 540-554Published by: The MIT PressStable URL: http://www.jstor.org/stable/2646650 .

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THE GENERALIZED DYNAMIC-FACTOR MODEL: IDENTIFICATION AND ESTIMATION

Mario Fomi, Marc Hallin, Marco Lippi, and Lucrezia Reichlin*

Abstract-This paper proposes a factor model with infinite dynamics and nonorthogonal idiosyncratic components. The model, which we call the generalized dynamic-factor model, is novel to the literature and generalizes the static approximate factor model of Chamberlain and Rothschild (1983), as well as the exact factor model 'a la Sargent and Sims (1977). We provide identification conditions, propose an estimator of the common components, prove convergence as both time and cross-sectional size go to infinity at appropriate rates, and present simulation results. We use our model to construct a coincident index for the European Union. Such index is defined as the common component of real GDP within a model including several macroeconomic variables for each European country.

I. Introduction

ECONOMIC activity in market economies is character- ized by phases of upturns followed by phases of

depression, which is manifested by the cyclical behavior and comovements of many macroeconomic variables. If comovements are strong, it makes sense to represent the state of the economy by an index-the reference cycle--describing the common behavior of such variables. This idea, first sug- gested by Burns and Mitchell (1946), is behind the NBER coincident indicator. The formal model that best captures it is the index model, or dynamic-factor model, proposed by Sargent and Sims (1977) and Geweke (1977). A vector of n time series is represented as the sum of two unobservable orthogonal components, a common component driven by few (fewer than n) common factors, and an idiosyncratic component driven by n idiosyncratic factors. If we have only one common factor affecting all of the time series only contemporaneously (that is, without lags), such a factor can be interpreted as the reference cycle (Stock & Watson, 1989).

Factor models can also be used to address different economic issues. For instance, a factor structure is often assumed in both financial and macroeconomic literature to estimate insurable risk. The latter is measured by the variance of the idiosyncratic component of asset prices (finance) or of output (macroeconomic risk sharing). More- over, factor models can be used to learn about macroeconomic behavior on the basis of disaggregated data (sectors, regions). (Quah and Sargent (1993), Forni and Reichlin (1996, 1997, 1998), and Forni and Lippi (1997) are useful

references.) Finally, factor models can be successfully used for prediction (Stock & Watson, 1998).

In the above examples, n-the number of cross-sectional units (different macro variables, returns on different assets, data disaggregated by sector or region)-is typically large, possibly larger than the number of observations (T) over time. VAR or VARMA models are not appropriate in this case, because they imply the estimation of too many parameters. Factor models are an interesting alternative in that they can provide a much more parsimonious parametri- zation. To address properly all the economic issues cited above, however, a factor model must have two characteris- tics. First, it must be dynamic, because business cycle questions are typically dynamic questions. Second, it must allow for cross-correlation among idiosyncratic components, because orthogonality is an unrealistic assumption for most applications.

The model we propose in this paper has both characteris- tics. It encompasses as a special case the approximate-factor model of Chamberlain (1983) and Chamberlain and Roth- schild (1983), which allows for correlated idiosyncratic components but is static. And it generalizes the factor model of Sargent and Sims (1977) and Geweke (1977), which is dynamic but has orthogonal idiosyncratic components.

An important feature of our model is that the common component is allowed to have an infinite moving average (MA) representation, so as to accommodate for both autoregressive (AR) and MA responses to common factors. In this respect, it is more general than a static-factor model in which lagged factors are introduced as additional static factors, because AR responses are ruled out in such a model.

The paper has three parts: population results, estimation, and empirics. In the population section, we show that the common and the idiosyncratic components are asymptotically identified. Moreover, we prove that, if we have q-dynamic factors, the first q-dynamic principal component series of the observable variables converge to the factor space as n - oo, and the projection of each variable on the leads and lags of these principal components converges to the common component of the variable.

The second part focuses on estimation. We propose an estimator of the common components which is the empirical (finite T) counterpart of the projection above. Building on the population results, we show that such an estimator converges to the common component as both n and T go to infinity. Simulation results show that our estimator performs well even when T is relatively small, possibly smaller than n.

In the empirical section, we use data on several macroeconomic variables for the countries of the European Monetary Union and compute a reference Euro-zone business cycle, which is defined as a weighted average of the common

Received for publication January 14, 1999. Revision accepted for publication December 14, 1999.

* Universit'a di Modena; Universite Libre de Bruxelles, ECARES, and ISRO; Universita di Roma; and Universit6 Libre de Bruxelles, ECARES, and ISRO, respectively.

This research has been supported by an A.R.C. contract of the Commu- naut6 fran,aise de Belgique and by the European Commission under the Training and Mobility of Researchers Programme (Contract No. ERBFMRXCT98-0213). We would like to thank Christine De Mol for generously giving us her time, Serena Ng for pointing out programming problems, Jim Stock for helpful advise, and Jorge Rodrigues for very valuable research assistance. The authors are grateful to an anonymous referee and to the Editor, James Stock, for pointing out an error in an earlier version of the paper.

The Review of Economics and Statistics, November 2000, 82(4): 540-554 ? 2000 by the President and Fellows of Harvard College and the Massachusetts Institute of Technology



THE GENERALIZED DYNAMIC-FACTOR MODEL: IDENTIFICATION AND ESTIMATION 541

components of the GDPs of the countries of the Union and can be driven by more than one common factor. On the basis of our results, we also evaluate the performance of variables (such as the sentiment indicator and the spread) that are usually taken as reference for the European business cycle.

This paper is closely related to three recent papers. Fomi and Lippi (1999) analyze the generalized dynamic-factor model proposed here from a purely theoretical point of view. They do not deal with estimation problems, but, unlike here, where we assume a factor structure from the start, they provide the conditions in population under which such structure exists. Fomi and Reichlin (1998) deal with estimation and empirics and show consistency of an estimator for the common component in a dynamic-factor model in which the idiosyncratic terms are mutually orthogonal. They also analyze identification of the common factors. Stock and Watson (1998) deal mainly with forecasting in a specification that is different from ours in that it allows for time-varying factor loadings but not for autoregressive dynamics.

II. The Model

We suppose that all the stochastic variables taken into consideration belong to the Hilbert space L2(fl, g P), where (fQ, .9 P) is a given probability space; thus, all first and second moments are finite. We will study a double sequence

ixit i E Ngt E }9

where

xit = bil(L)ult + bi2(L)u2t + * + biq(L)Uqt + tit, (1)

L standing for the lag operator, and suppose that assumptions (1) through (4) hold.

Assumption (1):

(I) The q-dimensional vector process {(ult u2t ... uqt), t E Z} is orthonormal white noise. That is, E(ujt) = 0; var (ujt) = 1 for any j and t; ujt L Ujt-k for any j, t, and k 0; Ujt I Us,t-k for any s # j, t, and k;

(II) g = {tit, i E N, t E Z} is a double sequence such that, firstly,

gn =l(tlt t2t . . . tnt) et E Z}

is a zero-mean stationary vector process for any n, and, secondly, tit I Uj,t1 for any i, j, t, and k;

(III) the filters bij(L) are one-sided in L and their coefficients are square summable.

Assumption 1 implies that the n-dimensional vector process Xn = {xnt, t E Z}, where

Xnt = (Xlt X2t ... Xn. 9

is zero-mean and stationary for any n. Trend-stationary processes can be easily treated with the tools developed below, which are applicable to the stationary residuals from deterministic detrending (while, in the case of difference stationary processes, our analysis can be applied to the result of differencing and mean subtracting).

The variables ujt, j = 1, ... , q, will be called the common shocks of model (1), the variables Xit = xit- it and tit will be called the common component and the idiosyncratic component of xit, respectively.

Model (1) is a factor analytic model. It is dynamic as the models employed in Geweke (1977) and Sargent and Sims (1977). However, here the cross-sectional dimension is infinite. This feature is the same as in the static-factor model of Chamberlain (1983) and Chamberlain and Rothschild (1983). An infinite cross section, together with assumptions (3) and (4) below, is crucial for the identification of our model. Indeed-and this is the third distinctive feature of model (1), which differentiates it from the dynamic-factor models mentioned above-we are not assuming mutual orthogonality of the idiosyncratic components tit. Without orthogonality, for fixed n, reasonable assumptions allowing for identification of the idiosyncratic and the common component would be very hard to find.

We do not assume rational lag distributions in equation (1). Through section III.A, we impose only a bounded spectral density for {xitl, for any i. In section III.B, further requirements, allowing for consistent estimation, will be introduced. We denote by ,n(O) the spectral density matrix of the vector process xnt and by -ij(O) its entries. (Note that the matrices ,n and I, n < m, are nested, so that no reference to n is necessary for oij(0).)

Assumption (2): For any i E 1N, there exists a real ci > 0 such that vii(0) ' ci for any 0 E [-ir, 7r].

Note that we are not assuming that boundedness of vii(O) is uniform in i. Note also that assumption (2) implies that all the entries uij(0) Of ,n(0) are bounded in modulus.

Now, denote by Xnj the function associating with any 0 E [-,t, NT] the real nonnegative jth eigenvalue of ln(0) in descending order of magnitude. The functions Xnj will be called the dynamic eigenvalues of In.- In the same way, with obvious notation, Xx and Xn denote the dynamic eigenvalues of Ix and V, respectively. The latter will be called common and idiosyncratic eigenvalues, respectively.

I We use the term dynamic eigenvalues to insist on the difference between the functions X and the eigenvalues of the variance-covariance matrix employed in the static principal component analysis. A standard reference for eigenvalues and eigenvectors of spectral density matrices is Brillinger (1981, chap. 9).



542 THE REVIEW OF ECONOMICS AND STATISTICS

Assumption (3): The first idiosyncratic dynamic eigenvalue XA is uniformly bounded. That is, there exists a real A such that X1(0) ? A for any 0 E [-fT, IT] and any n E R1.

Assumption (4): The first q common dynamic eigenvalues diverge almost everywhere in [ -IT, f]i. That is,

lim XX.(0) = oo for]j q, a.e. in [-IT, IT]. n- oo

Assumptions (3) and (4) call for some explanation. Assumption (3) is clearly satisfied if the x's are mutually orthogonal at any lead and lag and have uniformly bounded spectral densities, but is more general as it allows, so to speak, for a limited amount of dynamic cross-correlation. Similarly, assumption (4) guarantees a minimum amount of cross-correlation between the common components. With a slight oversimplification, assumption (4) implies that each uj1 is present in infinitely many cross-sectional units, with nondecreasing importance. (On assumption (4), see also remark (5) in section III.A.) On the contrary, assumption (3) implies that idiosyncratic causes of variation, although possibly shared by many (even all) units, have their effects concentrated on a finite number of them, and tending to zero as i tends to infinity. For example, assumption (3) is fulfilled if var (tit) = 1, cov (tit, ti? it) = p 0 0, while cov (it, ti+h,t) =

Oforh> 1. Note that in assumption (4) we require divergence "al-

most everywhere." The reason is twofold. Firstly, we do not need divergence everywhere to prove our results. Secondly, cases in which divergence does not hold everywhere can arise in very elementary situations. Suppose, for example, that xit = Ut + tit, where tit is nonstationary but (1 -L)tit is stationary. Then consider the variables (1 - L)xit = (1 - L)ut + (1 - L)t. Assuming that the variables (1 -L)it fulfill assumption (3), the model for the variables (1 -L)xit fulfills assumptions (1) through (4) with Xit = (1 - L) ut and XXI(0) = nl -e-i 0I2, which is divergent in [-IT, 'n] with the exception of 0 = 0.

Our first result is the following.

Proposition (1): Under assumptions (1) through (4), the first q eigenvalues of I diverge, as n - oo, a. e. in [-ir, ir], whereas the (q + l)th one is uniformly bounded. That is, there exists a real M such that n,q+1(0) ' M for any 0 E [-13, r] and any n E FN.

Proof: See the appendix.

The importance of proposition (1) lies in the fact that it transforms statements on the dynamic eigenvalues associated with the unobservable components Xn and gn into statements on the dynamic eigenvalues associated with xn, which is supposed observable. Moreover, as proved by Forni and Lippi (1999), the converse of proposition (1) also holds: if the first q eigenvalues of I,n diverge, as n - oo, a.e. in [-aT, ii], whereas the (q + i)th one is uniformly bounded, then the x's can be represented as in equation (1). Thus, if the

analysis of the dynamic eigenvalues of the observed process leads to the conclusion that the first q eigenvalues diverge a.e. in [-IT, iii, whereas the (q + l)th one is uniformly bounded, then the hypothesis of a model of the form (1) with q factors is plausible.

We call model (1), under assumptions (1) to (4), the generalized dynamic-factor model. We will show that, under assumptions (1) through (4), the common components Xit and the idiosyncratic components tit are identified and can be consistently estimated. On the other hand, it must be stressed that in this paper we do not deal with identification and estimation of the shocks ujt or the filters bij(L). Thus, we are not interested here in whether representation (1) has a structural interpretation or not.2 In this respect, even the assumption that the filters bij(L) are one-sided could be dropped with no consequence.

III. Recovering the Common Components

A. Population Results

In this section, our task is to construct an estimator of Xit, for any given i, based on the finite set of variables {xi, i = 1,.. ., n, t = 1, ... , T}, and to prove consistency for such an estimator as n and T tend to infinity. The proof is obtained in two steps. In the first step (III.A), we consider the projection of xit on all leads and lags of the first q-dynamic principal components (see the definition below) of xn, obtained from the population spectral density matrix in. We show that this projection, call it Xit,n, converges to Xit in mean square as n tends to infinity. In the second step (III.B), we construct the finite-sample counterpart of Xit,nq which is based on the estimated spectral density JT, call it Xit*n Then we combine convergence of Xit,n to Xit with the fact that XTn is a consistent estimator of Xit,n for any n as T tends to infinity, thus obtaining the desired result.

Let us recall that given the spectral density matrix Kn(O), there exist n vectors of complex-valued functions

Pnj(0) = (Pnj,l(0) Pnj,2 (0) ... *Pj,n(0A))

1= 1, 2, ... , n, such that (i) pnj(O) is a row eigenvector of Z1(O) corresponding to

Xnj(0); that is,

Pnj(O)n(O) = Xnj(O)Pnj(O) for any 0 E [-IT, 7r];

(ii) Ipnj(0) 12 = 1 for any j and 0 E [-uq, 7r];

(iii) Pnj MOns(0) = 0 for any j / s and any 0 E [-IT, 'iT];

(iv) pnj(0) is 0 - measurable on [ -T, IT];

where, as usual, we denote by D the adjoint (transposed, complex conjugate) of a matrix D. (For existence and properties of the functions pnj(0), see Brillinger (1981, ch. 9) and Fomi and Lippi (1999).)

2 On the identification and estimation of the common shocks in a related model, see Fomi and Reichlin (1998).




Any n-tuple fulfilling properties (i) through (iv) will be called a set of dynamic eigenvectors of In. Note that, apart from some inevitable complication, dynamic eigenvectors are nothing else than eigenvectors of the spectral density matrix, as functions of the frequency 0. A consequence of (ii) and (iv) is that dynamic eigenvectors can be expanded in Fourier series:

P J__ pr pn(0)ei0 d e-ikO

(this is the componentwise Fourier expansion of the vector pnj(()), where the series on the right side converges in mean square.

Defining

p~~1(L) = kp=-ik dO]L Pnj(L) 2 Jr p (0)e o Lk,

the filter pnj(L) is square summable. Moreover, assumption (2) implies that the scalar pnj(L)Xnl converges in mean square (Brockwell & Davis, 1987, p. 149, theorem 4.10.1). For j =

1, ... , n, the scalar process Ipnj(L)x,1t, t E Z}, whose spectral density is

PnJ((0)In((0)PnJ((0) = X nj(0)9

will be called the ith dynamic principal component of xn. A consequence of (iii) is that, if j / k, then the ith anid kth principal components are orthogonal at any lead and lag.

Now consider the minimal closed subspace of L2(Q, Y P) containing the first q principal components

?6K = span (p #(L)x,3t,

j = 1, ... , q, t E E),

and the orthogonal projections

Xit,n= proj (xit I Wn)-

We can obtain an explicit formula both for Xit,nz and the residual ;it,n = Xit- Xit,z by observing that

In = fni(()Pni(() + Pn2(0)Pn2(0) + * * + Pnn(O)Pnn(0).

(The vectors ptj(O) are an orthonormal system of eigenvectors for In.) Therefore,

xnt = Pn (L)Pn1(L)Xnt + Pn2(L)Pn2(L)Xnt

+ + Pnn(L)Pnn(L)Xnt

Taking the ith coordinate,

xit = [pn1,i(L)Pn1(L)Xnt + fin2,J(L)Pn2(L)Xnt

+ *+ nq,i(L)Pnq(L)Xnt] + [13n,q+?,i(L)Pn,q+?(L)xnt

+ *** + ~Pnn,i(L)Pnn(L)Xnt].

Now, since the dynamic principal components are mutually orthogonal at any lead and lag,

Xit,n = Kni(L)Xnt, (2)

with

Kni(O) = Pin1,U(O)P00(O) + Iln2,J(0)Pn2(0)

+ *.. + finq,i(0)Pnq(0)

Remark (1): Note that in equation (2) the orthogonal projection Xit,n is expressed as the sum of the orthogonal projections of xit on (leads and lags of) each of the first q dynamic principal components; that the coefficients of the ith orthogonal projection are the coefficients of the filter

pnj(L); and that, obviously, analogous formulae and observations hold for (it,n and the principal components from q + I ton.

Let us now state and comment our first step toward recovenng Xit.

Proposition (2): Suppose that assumptions (1) through (4) hold. Then,

lim Xit,n = Xit n---+0

in mean square for any i and t.


Remark (2): Note firstly that Xit,n, that is, the population approximate common component of xit, results from a simple rule involving the dynamic eigenvectors of the matrices n, with no intervention of the unobservable x's and c's. Thus, we are ready for the second step, in which we construct an empirical approximate common component based on the observable xnt, for t = 1, . . . , T.

Remark (3): An intuitive insight into proposition (2) can be obtained by considering the following example:

Xit = Ut + tit, (3)

where all i's are white noise, have unit variance, and are mutually orthogonal at any lead and lag. In this one-factor case pnI(L) = (1/ / iF . .. 1/), so that

Xit,n = 5n ,i(L)PnA)Xnt= (1/n 1/n ... I/n)xnt

I n - ut + - z u. n s=1

Convergence of Xit,n to Xit in mean square thus follows from var (In1=IsIn) = 1/n. In this example, the filter

j3l,i(L)pD1(L) is nothing else than the standard arithmetic




mean of xnt,. In the appendix, we show that in general the filters p-n j,A(L)pn(L), for j = 1, . . . , q, which average the x's both over the cross section and over time, share with the standard arithmetic mean the property that the sum of the squared coefficients tends to zero as n tends to infinity. Assumption (3) indeed ensures that P-nj,i(L)Pni(L)9nt vanishes as n tends to infinity (see the appendix), so that, because

nj,i(L)Pnj(L)Xnt = finj,i(L)Pnj(L)Xnt + Pnj,i(L)Pnj(L)9nt

in the limit only the term 5nji(L)Pnj(L)Xnt survives. How- ever, proving that in general >1q_1flnj,i(L)pnj(L)Xnt converges to Xit is not as elementary as in model (3).

Remark (4): Assume again, for simplicity, that q = 1 but that the model is general: xit = bi(L)ut + (it. Now suppose that we take the standard arithmetic mean xnt of xnt, instead of the first dynamic principal component and that we project xit on all leads and lags of nt. Call -it,n this projection. Assumption (3) ensures that the idiosyncratic part of Xnt tends to zero, so that the projection Xit,n tends to the projection of xit on the space spanned by the common components (Xit). This estimation method can be extended to q > 1 by using q averages with different systems of weights, as in Forni and Reichlin (1998). An advantage of their method is that the coefficients of their averages are independent of the x's and not estimated (as in our case). However, unless ad hoc assumptions are introduced, near singularity of the chosen averages for n growing, with the consequence of inaccurate estimation, cannot be excluded. This problem is completely solved with dynamic principal components, which are mutually orthogonal at any lead and lag.

Because Xit,n depends only on xnt, proposition (2) has the immediate implication that the components Xit and tit are identified. More precisely, we can state the following corollary.

Corollary (1): Suppose that xit can be represented as in equation (1), and that assumptions (1) through (4) are fulfilled. Suppose that xit admits the alternative representation

xit = bi1(L)u'It + bi2(L) I2 + * + bi4(L)Ui4t + (it, (4)

and that assumptions (1) through (4) are also fulfilled for equation (4). Then, Xit = Xit, so that tit = tit. Moreover q4 = q.

Remark (5): An important consequence of corollary (1) is that representation (1) is nonredundant; that is, no other representation fulfilling assumptions (1) through (4) is possible with a smaller number of factors. In the following example, we have a common-idiosyncratic representation of the form (1) with one factor. However, because assumption

(4) is not fulfilled, another representation with zero factors fulfilling assumptions (1) through (4) is possible. Specify equation (1) as

xit= b ut + tit,

where t is defined as in model (3). Now suppose that the sequence of coefficients bi, i E RJ, is square summable (that is 'i1b 2 < oo). In this case, as the reader can easily check, the first eigenvalue of 1,(O) is 1 + EI'lb 2, and is therefore bounded as n tends to infinity. Thus, the x's-though the correlation between xit and xjt never vanishes-are purely idiosyncratic.

Naturally, in empirical situations we do not know the number q. However, another implication of proposition (2) is that assuming a q* larger than the actual q has no dramatic consequences, because the expected mean-squared difference between the resulting projections X*it,n and X*,n averaged over the cross-sectional units, is asymptotically zero. Precisely:

Corollary (2): Under assumptions (1) through (4), let Xitn be the projection of xit on the space spanned by all leads and lags of thefirst q * dynamic principal components, with q * > q. Then

I n

lim - E[(X* *,-Xit,n )2] = ?. n-o n i i1


A dynamic-factor model with an infinite cross-sectional dimension is studied by Stock and Watson (1998). Among several differences, let us observe here that their model is more general than ours in that their factor-loading coefficients are allowed to be time varying. On the other hand, in Stock and Watson's paper, the common components are modeled (in our notation and assuming for simplicity only one factor) as ci(L)c(L)u, with polynomials ci(L) of finite order, which is dynamically more restrictive than equation (1). Stock and Watson construct estimated factors that converge to the space spanned by the "true" factors. This corresponds, in this paper, to the statement that the estimated counterparts of Pnj(L)xnt converge to the space spanned by the x's (or the u's). In this paper, we prove this result and go a step further, showing that the estimated Xit,n converges to Xit for any i. (See the comment under lemma (4), appendix.)

B. Estimation Results

Proposition (2) shows that the common component Xit can be recovered asymptotically from the sequence Kni(L)Xnt. The filters Knj(L) are obtained as functions of the spectral density matrices ln(O). Now, in practice, the population spectral densities ln(O) must be replaced by their empirical




counterparts based on finite realizations of the form

Xn= (xn1 Xn2 * XnT).

On the other hand, consistent estimation of the spectral density requires a strengthening of assumption (2). Pre- cisely, we replace assumption (2) by

Assumption (2'): The vector xn, has a representation

x Xnt = I CkZt-k,

k= -oo

where Zt is an n-dimensional white noise with non-singular covariance matrix and fourth-order moments, and

a ICij,kllkl 12 < X, k= -oo

for i, j=1,.. , n, where Cijk is the i, j entryof Ck.

Under assumption (2'), if nT(0) denotes any periodo- gram-smoothing or lag-window estimator of n(0), based on xT, we have

lim P[ sup o1(O) - u1(O)I > E] = 0, (5) T-X-0 0EJ-,nu]

where UT(0) denotes the i, j entry of NT(0). (See Brockwell and Davis (1987, p. 433).) Under assumption (2'), the estimated counterpart of Kni(O) allows for a consistent reconstruction of the factor space. More precisely, we prove that the projection of xit onto the space spanned by the first q empirical principal components converges to the common component Xit.

Denote by XT (0) and pnT(0), respectively, the eigenvalues and eigenvectors of the matrix Xn'(O). Since eigenvalues and eigenvectors are continuous functions of the entries of the corresponding matrix, convergence (5) implies that XAT(0) and pn(O) converge to Xnj(O) and Pnj(O)g respectively, in probability, uniformly in 0 E [-q, a'], for T o oo. More- over, considering

K T(O) = fiT1,O)p (O) + pTi~2(O)pT (O) Kni0 Pnl i(0)PnI(0 +n2,i )n2(

+ *** + Pn'q,(O)Pn'q(n)q

that is, the empirical counterpart of Kni(O), i < q, KnT(0) converges to Kni(O) in probability, uniformly in 0 E

[-mg, i], for T - oo. Thus, for all E > 0 and - > 0, there exists T1 = T1 (n, E, q) such that, for all T ' T1,

P[ sup |K T(0) -Kni(0)I > E] ` 1i. (6)

Now, observe that, in principle, given the estimated spectral density matrix JT(0), K T (0) can be computed for

any 0, so that each of the coefficients of the corresponding two-sided filter

ni(L)= , KnikLk xr K2Tk=Tn i

can be obtained. However, in practice, the projection K T (L)Xnt of xi, onto the space spanned by the first q empirical principal components cannot be computed, because, for t ' 0 and t > T, Xn, is not available. Therefore, a truncated version of the estimated filters K T (L), of the form Ek--M(T)Ki kLk, where M(T) oo is such that lim SUpT 0M3(T)IT < ?o as T - oo, is considered. This T-113 rate for the window width M(T) is related to the method of proof (see the proof of Proposition (3) in the Appendix), and is probably not essential to the consistency result itself. Even this truncated version of K T.(L) however has to be truncated further when acting on Xn,, t < M(T) or t > T -M(T), yielding

min(t- 1,M(T))

Kni(L)= ! Kni, Lk.

k=max(T-t.-M(T))

Due to this unavoidable truncation, the common component Xit, for fixed t, never can be recovered, even as n and T tend to infinity. Indeed, part of its variance is lost because of the non-observability of Xnl, t ' 0 and t > T. We therefore restrict our attention to the "central part" of the observed series, the values of t of the form t = t*(T), with

t*(T) t*(T) O<a?liminf -limsup b <1. (7)

T-X'z T T-Xo T

The following result then provides the empirical counterpart of proposition (2).

Proposition (3): Assume that assumptions (1), (2'), (3), and (4) are satisfied. Then, for all E > 0 and - > 0, there exists No(E, -q) such that

[ |K Tt(L)Xnt -Xit I > E] C5 'q

for all t = t*(T) satisfying equation (7), all n ' No and all T larger than some To(n, E, 1q).

Proof: Throughout, we write aT instead of [aTI (the smallest integer larger than or equal to aT), and bT for [bTl (the largest integer smaller than or equal to bT). We also tacitly assume that T is large enough for M(T) being strictly less than min (aT, T - bT + 1, (bT - aT)12).

For any t E [aT, bT], KnT(L) then reduces to

-M(T)Kni Lk We have k= nT),k

P [|KT(L)xnt -Xitl > E] C P[|(KTt(L) - Kli(L))xntI > Ed2]

+ P[|Kn(L)xnt -Xit > E/2] = RTt + R,32, say.




Proposition (2) ensures the existence of an NO(E, -q) such that, for n ' No Rn2 ? . As for RT, we have, from the definition of KnT2n

M(T)

nl P T (Knih - Kni,h)Lh

Lh =-M(T)

-M(T)- 1 ?? E' - al Kni,hL h- a Kni,hLh Xnt >

h=-oo h=M(T)+l 2J

? P [ ) I (Knih - Kni,h)LhXnt > ] h = -M(T)

-M(T)-1 0I

+ P Kni,hLh - Kni,hLh Xnt >4 1 kh=-oo h=M(T)+ 1 4h

=R Tt +R T -nli nl2-

Since Ki(L) is a nonrandom square-summable filter, there

exists T2(n, E, -q) such that R T 2 ? 4 for all T ' T2 and aT <

t < bT. Turning to R,Ttl it follows from Chebyshev's theorem and equation (6) that, for T ? T1 (n, 8, 8)

RT C~ [ ( M(T) (KnTih - Kni,h)L' xnt > 4

and os[up ] Kn(O) -Kni(O)l ? 6]

+ P E[u supI Kni(O) - Kni(O) |>6]1

16 [M(T) 2

-?2 E h (Knih - Kni,h)L xnt h =-M(T)

Xn supIT() ni ~ ni()I-

X P sup IK T i(0) - Kn( - 8

If the filter K T* and the observation xnt were independent, then, in view of the classical properties of dynamic principal components (see the proof of Proposition 1 in the appendix), we would have (denoting by 1'n,h = E[Xfl,tXn,t-h], h = 0, +1, . . . the autocovariance function of Ixnt1)

M(T) 2

h - (KE i - K TlK,h)L xnt h=-M(T)

X sup K Kni T)- K- K(O) ?-E)]

[M(T) M(T)

=E E= r (Knik- Knisk)Fnllk k= -M(T) I= -M(T)

- (KlIKni,1) |sup |KnT(O) -K_(0) <6)]

< Eil6e2 f Tan(0) do] = 62 f nt w(0) dO.

_ ni

Thus, letting 82 = E2-q/128 fIs7r nI(O) dO, for n ' NO(E, -q)

and T ' max (Ti(n, 8, 8), T2(n, E, -q)), we would obtain RnT ? "hence RnT + RnT2 < q for all t = t*(T) satisfying condition (7). The proof of Proposition 3 then would be complete-without any rate assumption on M(T).

Unfortunately, Kni and xnt are not independent. For Tlarge enough, they are "almost independent," though, so that the above reasoning is essentially correct; moreover, it provides the right insight into the intuitive ideas underlying the proof. A more formal treatment, taking into account the non- independence between K?T and Xnt, is given in the Appendix.

IV. The Proposed Estimator and the Choice of q

In light of the results of the previous section, we propose the following estimator. For some selected integer M = M(T), we compute the sample covariance matrix rFT of X,t and Xn,t-k for k = 0, 1, ... , M and the (2M + 1) points discrete Fourier transform of the truncated two-sided se-

quencern,-M, . . .,9 rnO' ... 9. rnM where rn,-k = r'nk More precisely, we compute

M

InT(Oh) = n Fnk(Oke (8) k= -M

where

oh = 2lhl(2M + 1), h=O, 1, ..., 2M,

and Wk = 1 - [Ik/(M + 1)] are the weights corresponding to the Bartlett lag window of size M. Consistent estimation of ln(O) (which is required for the validity of proposition (3)) is ensured, provided that M(T) -

oo and M(T)IT - 0 as T ) oo.

Then we compute the first q eigenvectors pnTj(Oh), = 1, 2, ... , q, of OT()h), for h = 0, 1, ... , 2M.3 Finally, for h = 0, 1, ... , 2M, we construct

KnT(Oh) =

PIT ,i(Oh)PT1(Oh) + * * + Tqji(Oh)PnTq(Oh).

The proposed estimator of the filter Knj(L), j = 1, 2, ... , q, is obtained by the inverse discrete Fourier transform of the vector

(K (T0) * ... 9 K,T( 2M)),

that is, by the computation of

1 2M

Kni,k 2M + 1 h= ni(Oh)e

Note that, for M = 0, pnT(00) is simply the jih eigenvector of the (estimated) variance-covariance matrix of xnt: the dynamic principal components then reduce to the static principal components.




for k = -M, . . , M. The estimator of the filter is given by

M

(L)- K, Lk. (9) k= -M

Note that the same integer M has been used as the size of the Bartlett window in the estimation of T,(0), and as the truncation length of KT!(L), so that imposing M(T) =

O(T113) ensures both consistency of the estimated spectrum and consistency of the estimated common component (see Proposition 3). In particular, it appears that M(T) = round (2T1/3) performs remarkably well in the simulations reported in Section V. As an alternative, we could take any sequence MO(T) such that MO(T) ) oo as T - oo, and MO(T) = O(T113), estimate all of the specifications with 0 ? M ' MO(T), and choose the one minimizing some dynamic specification criterion. Although a data-dependent rule seems preferable in principle, we found that the standard AIC and BIC criteria underestimate the optimal lag-window size, so this topic is left for further research.

So far, we have assumed that q, the number of nonredundant common factors, is known. In practice of course, q is not predetermined and also has to be selected from the data. Proposition (1) can be used to this end, because it links the number of factors in equation (1) to the eigenvalues of the spectral density matrix of x,: precisely, if the number of factors is q and g is idiosyncratic, then the first q dynamic eigenvalues of Y.,(O) diverge a.e. in [--rr, rr] whereas the (q + l)th one is uniformly bounded.

However, no formal testing procedure can be expected for selecting the number q of factors in finite-sample situations. Even letting T D oo does not help much. The definition of the idiosyncratic component indeed is of an asymptotic nature, where asymptotics are taken as n D oo, and there is no way a slowly diverging sequence (divergence, under the model, can be arbitrarily slow) can be told from an eventually bounded sequence (for which the bound can be arbitrarily large). Practitioners thus have to rely on a heuristic inspec- tion of the eigenvalues against the number n of series.

More precisely, if T observations are available for a large number n of variables xi,, the spectral density matrices ,T, r ' n, can be estimated, and the resulting empirical dynamic eigenvalues XT) computed for a grid of frequencies. The following two features of the eigenvalues computed from

r T,r = 1, . .. , n, should be considered as reasonable evidence that the data have been generated by equation (1), with q factors.

1. The average over 0 of the first q empirical eigenvalues diverges, whereas the average of the (q + l)th one is relatively stable.

2. Taking r = n, there is a substantial gap between the variance explained by the qth principal component and the variance explained by (q + 1 )th one. A preassigned minimum, such as 5%, for the explained variance,

To illustrate the use of criteria (1) and (2), we have generated data from a two-factor model (model M4 below) with n = 50 and T = 100. Then, we have estimated the spectral density matrix for a grid of frequencies, using equation (8) with M = 10. Lastly, we have computed the eigenvalues of the upper-left r X r submatrices, r =

1, . .. , n. Figure 1 reports the plot of the averages over frequencies

of the theoretical and estimated eigenvalues. On the horizontal axis, we indicate the number of cross-sectional units r, which obviously is maximum when the whole sample n =

50 is considered. Features (a) and (b) emerge quite clearly: the first q averaged eigenvalues exhibit an approximately constant positive slope, while the remaining ones are rather flat; moreover, the variance explained by the qth principal component is substantially larger than the variance explained by the (q + l)th one, even for small r.

To conclude this section, let us remark that, when applying criteria (1) and (2), we should keep in mind that, as indicated by corollary (2), setting a number q* of factors larger than the true one q cannot have dramatic consequences on estimation.

V. Simulation Results

In order to evaluate the performance of our estimation procedure for finite values of n and T, we have carried out Monte Carlo experiments on the following four two-factor models.

FIGURE 1 -DYNAMIC EIGENVALUES AVERAGED OVER FREQUENCIES, MODEL M4

180

160 -

140 -

120

100 X

80

60-

40-

20 -

0 5 10 15 20 25 30 35 40 45 50

Horizontal axis: r; vertical axis: variance/2Tr.

could be used as a practical criterion for the determina- tion of the number of common factors to be retained. This 5% limit is used in the empirical exercise of section VI.




Static model:

xit = aiult + bjU2t + 2it (M1)

Static with delay:

xit = ajul + biu2t + +2git for i even (M2)

xit = aiult-I + biU2t-I + 2it for i odd.

MA(1) common component:

xit = a0iult + aliult-I + bOjU2t + bliu2t_i + 2git. (M3)

AR(1) common component:

al b Xit =- U1t + U2t + 25tit. (M4) 1-c-L 1-diL

In all these models, ult, U2t, ai, a0i, ali, bi, boi, bli, and git are i.i.d. standard normal deviates, while ci and di are uniformly distributed over [-0.8, 0.8], in order to ensure costationarity of the x's. Note that the idiosyncratic shocks are multiplied by a constant so that, on the average, all cross-sectional units have common-idiosyncratic variance ratio 1, in all models.

We generated data from each model with n = 10, 20, 50, 100, and T = 20, 50, 100, 200, and applied the estimation procedure described in section IV with M(T) = round

[2T1/3]. Each experiment was replicated 400 times. We measured the performance of our estimator, ît, by

means of the criterion

I ( ît - Xt)2 i,t

R(X, X) = 2

xit

i,t

Table 1 reports the average and the standard deviation (in brackets) of this statistic across the experiments.

For all models, we see that the fit improves as both n and T increase. To better appreciate the results, we add a row reporting R(3, X), where -it is the infeasible estimate of the common components obtained by performing OLS regres- sions of the variables on the contemporaneous and lagged values of the unobservable true common factors ujt; Xit is computed only for n = 100. The AIC criterion is used for the choice of the number of lags. Note that, for the autoregressive model M4, the results obtained with n ' 50 are similar to those obtained with the true factors or even better, indicating that the error involved in approximating the factor space is negligible as compared with the error arising from the MA approximation of the AR dynamic structure implied by the OLS strategy.

VI. A Coincident Indicator for the EURO Currency Area

In this section, we use our method to compute a coincident indicator for the countries of the European Monetary Union. We estimate the generalized-factor model, using a large panel including several macroeconomic variables for each EURO country. The coincident indicator is constructed as the weighted average of the common components of countries' GDPs.

Our approach is similar in spirit to Stock and Watson (1989), who define the reference cycle as an unobserved index, common to many macroeconomic variables. How- ever, one important difference is that we allow for the possibility that more than one single common shock capture the comovements of the macroeconomic variables of inter- est. This is relevant whenever there is more than one source of aggregate fluctuations.

We proceed as follows. Step (1): We construct a panel pooling seven quarterly

macroeconomic indicators for all countries of the EURO zone, excluding Luxembourg, from 1985 to 1996. (See table 2, with X and - indicating, respectively, whether the series is available or missing.) Data are taken in logs and differ- enced (except for the spread, which is not transformed, and the sentiment indicator, which is simply taken in logs), and normalized dividing by the standard deviation. Thus, n = 63 andT= 51.

Step (2): We estimate the spectral density matrix, compute the dynamic eigenvalues, and identify q = 3, using criterion (2) of section IV.

TABLE 1.-AVERAGE AND STANDARD DEVIATION (IN BRACKETS) OF R(X,X) ACROSS 400 EXPERIMENTS

T= 20 T= 50 T= 100 T= 200

Model MI n = 10 0.554 (0.281) 0.394(0.201) 0.343 (0.162) 0.296 (0.131) n = 20 0.372 (0.174) 0.244 (0.091) 0.194 (0.068) 0.162 (0.045) n = 50 0.261 (0.098) 0.150 (0.035) 0.109 (0.024) 0.081 (0.014) n = 100 0.227 (0.069) 0.123 (0.024) 0.084 (0.014) 0.059 (0.008) R(X, X) with

n = 100 0.197 (0.105) 0.061 (0.012) 0.030 (0.005) 0.015 (0.002) Model M2

n = 10 0.671 (0.351) 0.472 (0.206) 0.382 (0.187) 0.317 (0.138) n = 20 0.505 (0.202) 0.295 (0.100) 0.070 (0.081) 0.047 (0.063) n = 50 0.390 (0.117) 0.048 (0.085) 0.026 (0.049) 0.016 (0.032) n = 100 0.353 (0.098) 0.032 (0.071) 0.016 (0.041) 0.009 (0.027) R(X, X) with

n = 100 0.366 (0.113) 0.105 (0.019) 0.052 (0.008) 0.025 (0.004) Model M3

n = 10 0.633 (0.255) 0.436 (0.152) 0.340 (0.108) 0.294 (0.081) n = 20 0.479 (0.160) 0.289 (0.084) 0.211 (0.051) 0.161 (0.029) n = 50 0.384 (0.106) 0.193 (0.039) 0.128 (0.022) 0.092 (0.013) n = 100 0.344 (0.084) 0.163 (0.029) 0.103 (0.014) 0.067 (0.007) R(X, X) with

n = 100 0.366 (0.112) 0.103 (0.019) 0.051 (0.007) 0.025 (0.003) Model M4

n = 10 0.642 (0.360) 0.433 (0.213) 0.352 (0.170) 0.299 (0.132) n = 20 0.459 (0.187) 0.278 (0.093) 0.201 (0.060) 0.166 (0.047) n = 50 0.342 (0.100) 0.193 (0.039) 0.131 (0.025) 0.095 (0.015) n = 100 0.322 (0.083) 0.167 (0.028) 0.108 (0.015) 0.073 (0.008) R(X, X) with

n = 100 0.424 (0.118) 0.199 (0.028) 0.114 (0.015) 0.067 (0.007)




Step (3): We estimate the common component of GDP for each separate country, following the procedure of section IV. Figure 2 reports the resulting estimates.

Step (4): We construct the weighted average of the common components above using as weights the GDP levels. This is the proposed coincident indicator. We illustrate it in figure 3.

Some remarks are needed. Firstly, results from step (2) show that the presence of a cycle in the strong sense of Stock and Watson (1989)-that is, a single common factor-is not supported by this data set.

Secondly, in our methodology, output plays a prominent role. Because the data do not support a single static factor, the cycle must be defined as the common component of a particular cross-sectional unit. Clearly, GDP is the most natural choice as the reference variable. On the other hand,

we are interested in the common component of output and not in output itself, because we want to disregard that part of GDP variation that is poorly correlated with other variables. Hence, the latter also play an indirect role in the construction of the index, through the estimated dynamic principal components.

Finally, with our methodology there is no need to distinguish a priori between leading and coincident variables. The weight of each variable in the index depends on the cross-correlations at all leads and lags: a variable which leads with respect to GDP, for example, will have small contemporaneous weight and will be shifted automatically in the appropriate way.

In order to understand better the structure of the multicoun- try, multivariate correlations, we also compute the common components of variables other than GDP and construct two sets of statistics. First, for each variable, we compute the ratio of the variance of the common component to total variance, for each country and for the aggregate EURO area (table 3). These ratios measure the degree of commonality of each variable in the system. Second, we compute, for each variable, the average contemporaneous correlation coeffi-

TABLE 2.-THE DATA

Countries GDP Cons. Inv. CPI Spread Sent. I.P.

Germany X X X X X X X France X X X X X X X Italy X X X X X X X Netherlands X X X X X X X Ireland X X X X Spain X X X X X X X Finland X X X X X X Austria X X X X X X Belgium X X X X X Portugal X X X X X X X

GDP: GDP, s.a., in national currency, at constant (1990) prices. Source: OECD; for Germany and Portugal: IMF.

Cons.: Private final consumption expenditure, s.a., in national currency at constant (1990) prices. Source: OECD; for Germany and Portugal: IMF.

Inv.: Gross fixed-capital formation, s.a., in national currency at constant (1990) prices. Source: OECD; for Germany and Portugal: IMF.

CPI: Consumer Price Index, base year 1990. Spread: Difference between the government bond yield and the Treasury Bill rate (or the money market

rate depending on data availability), in percentage per year. Source: IMF. Sent.: economic sentiment indicator. Source: European Commission, DG II. I.P.: Industrial production, s.a., index number, base year 1990. Source: IMF.

FIGURE 2.-COMMON COMPONENT OF GDP OF NINE COUNTRIES

OF THE EURO ZONE

0.02 i * - Sr 0.02 F ra 0.02.Ita

0.01A 0.01 -O .01

86 88 90 92 94 96 96 66 66 90 92 94 96 96 86 88 90 92 94 96 98

0 .02 [.. 0 O .02 *Spa 0 .02 .l...J

-0.01 ......... 001 .01

-O 01 -O ~~~~~~~~~~~~~~~~~~~~~~~~001 -0.01Fi

86 88 90 92 94 96 98 86 88 90 92 94 96 98 86 88 90 92 94 96 98

0 02 Aut 0 .02 lel 0 04

0.01 0.01- 0.02

0 0 0

-0.01 -0.01 02

86 8 8 9 0 92 9'4 9 6 98 86 88 90 92 94 96 98 86 88 90 92 94 96 98

Horizontal axis: time. Solid line: common component of national GDP. Dotted line: common component of aggregate EURO-zone GDP (vertical scale for Portugal is not the same as for all other countries).

FIGuRE 3.-COINCIDENT INDICATOR FOR THE EURO ZONE

x 10-3 Coincident indicator for the EURO zone

1 5 _ . . . . . . . . . . . . . . . . . ... . . . . . . . . . ............_

s '10 .. t . ..... .. .i ..... . -

o ..... \ . . . .~~~. ........

0 . . . . . . . . . . .

- 5 - . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ... . . . . . . . . . ... . . . . . . . . . . . . . . . . . . . . . .

86 88 90 92 94 96 98

Honzontal axis: time.

TABLE 3.-PERCENTAGE OF VARIANCE EXPLAINED BY THE COMMON

COMPONENT


EURO aggregate 85 70 57 74 95 99 80 Germany 68 56 49 69 95 95 68 France 70 43 72 38 96 94 56 Italy 54 65 71 42 69 98 29 Netherlands 35 57 29 57 96 65 27 Ireland 39 51 87 22 Spain 96 73 96 37 34 68 62 Finland 46 65 90 46 Austria 55 99 53 Belgium 55 55 82 97 44 Portugal 54 43 54 69 63 69




cient with the common components of the other variables of the same country, for each country and the aggregate EURO area (table 4). These statistics measure the degree of synchronization of each variable with the other variables of the same country. Through these results, we can also evaluate the performance of variables that are typically used to describe the state of the economy, such as the sentiment indicator and the spread, and validate ex post the choice of GDP as the reference variable for the European cycle.

There are a few interesting findings. First, the common component of GDP has the largest average contemporaneous correlation for almost all countries and for the aggregate. This fact provides an ex post confirmation of our choice of the GDP as the reference variable for the coincident index. Note, however, that using directly the GDP, rather than the common component of GDP, as the index would not be a good choice, due to the presence of an idiosyncratic component that accounts for 15% of total variance.

Second, for most countries and for the aggregate, the sentiment indicator has the largest common component. However, its synchronization with the other variables is lower than that of GDP, which suggests that the sentiment indicator is not an appropriate coincident index, probably due to its leading behavior.

Note that the correlations between the common components appearing in table 4 are, in general, unexpectedly small. This is mainly due to the fact that, somewhat surprisingly, the inflation rate has very low or even negative synchronization.

VII. Summary and Discussion

The generalized dynamic-factor model analyzed in this paper is novel to the literature, in that it allows for both a dynamic representation of the common component and nonorthogonal idiosyncratic components. We have shown that, although for a finite cross-sectional dimension this model is not identified, identification of the common and the idiosyncratic components is obtained asymptotically as the cross-sectional dimension goes to infinity.

Because the idiosyncratic components are correlated, the model cannot be estimated on the basis of traditional

methods. We have proposed a new method, yielding consistent estimates of the components as both the cross section and the time dimensions go to infinity at some rate. More precise information on these rates would be interesting; however, such information typically would require much heavier assumptions on the heterogeneity of cross-sectional units. This is a topic of the authors' ongoing research; see Forni, Hallin, Lippi, & Reichlin (2000). The common components are computed as the projections of the observations onto the leads and lags of the dynamic principal components of the observations and the idiosyncratic components are derived as the orthogonal residuals.

The method is applied to a panel including several macroeconomic indicators for each of the EURO countries, in order to obtain an index describing the state of the economy in the EURO area. The European coincident indicator is defined as the common component of the European GDP.

TABLE 4.-AVERAGE CORRELATION OF THE COMMON COMPONENT WITH THE

COMMON COMPONENTS OF THE OTHER VARIABLES OF THE SAME COUNTRY


EURO aggregate 0.58 0.36 0.55 -0.20 0.14 0.41 0.51 Germany 0.55 0.33 0.49 -0.22 0.01 0.34 0.50 France 0.66 0.47 0.68 0.17 0.39 0.56 0.58 Italy 0.49 0.48 0.49 0.09 -0.52 0.46 0.34 Netherlands 0.33 0.09 0.15 0.07 0.26 0.31 0.01 Ireland - -0.11 0.44 0.36 0.41 Spain 0.64 0.62 0.63 0.09 0.17 0.55 0.36 Finland 0.35 - -0.21 - 0.40 0.04 Austria 0.47 - - 0.22 - 0.58 Belgium 0.50 - -0.08 0.30 0.31 0.43 Portugal 0.22 0.50 0.29 0.37 0.32 0.53

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APPENDIX A

Proof of Proposition (1). We need the following result (see Brillinger, 1981, p. 84, Exercise 3.10.16): Let A be an n X n, complex, hermitian nonnegative definite matrix, and let Xk, k = 1, . . ., n, be its (real) eigenvalues in descending order of magnitude. Denote by Db an n X (k - 1) complex matrixfor 1 < k s n, the n X 1 null matrix for k = 1. The eigenvalue Xk is the solution of

min max bAb Dk b (10)

s.t. |b| = 1, bDk = ?

Note thatfor k = 1 the only constraint is lb = 1. Since In =

IXI + In, given Dj, for bD- = 0 and b| = 1, then

max bln(4)b ? max blx(O)b,

mnax bln(O)b s maxb b (n)b + max bn(e)b

<max bl x(0)b + XA,I (0). b n i

From equation (10) and the above inequalities, we get

(a) Xnj(0) s AnXj(e) + xtl(e) and (b) X,l1(e) ? A\Xy(0)

The statement on the first q eigenvalues of I,n follows from equation (b). The statement on the (q + l)th one follows from equation (a) and the fact that the (q + l)th eigenvalue of Ix vanishes at any frequency. QED

To prove proposition (2) we need some intermediate results. We suppose that assumptions (1) through (4) hold.

Lemma (1): Denote by PnjJ,(0) the ith component of p,j(0), as defined in Section III.A. For j s q, limn-o IPnjJ (0) I = 0 a.e. in [-Tr, Tr].

Proof: Let_P,n be the n X n matrix having the eigenvectors Pn3 on the rows. From Pn diag (X,, I X,2 ... X,,n) P,i = ,In, one obtains

q r

Pn j,i(0) 12,() + I Pnj,i(0) 12X,n(0) = Ui(O), j=1 j=q+l

where oi is the spectral density of xit. By proposition (1), X,j(e) diverges a.e. in [-rr, 7r] for j - q. But ui(0) is a.e. finite in [-rr, -r]. QED

Lemma (2): For given i and n E FN, consider the n-dimensional filters (defined in section MIIA)

q

Kni(L) = z fi,jj(L)pnj(L). j=-1 -

Then

lim iJf K,l(0)|2de = 0,

where IKJ(6) 12 = Kn,(O)kijo)

Proof: We have

q

IK,l,(0)12 = I Pn1,i(e)12 s 1. j=1

Moreover, by lemma (1), I K,i(0) 12 tends to zero a.e. in [-nr, 7r]. The result follows from applying the Lebesgue dominated convergence theorem (Apostol, 1974, p. 270). QED

Lemma (3): For n E N let a,,(L) be an n-dimensional two-sided square-summablefilter Assume that

limfT I a,,(0) 1 2de = 0.

Then

lim a, (L)gnt = 0 n boo

in mean square.

Proof: From the same argument as in the proof of proposition (1), we have

var (a, (L)g,t)=f a,,(O)tl(O)An(0) de s fr xt(0) I a,(0)12 de.

The result follows from assumption (3). QED Denoting by 4,,i(O) the spectral density of Kni(L)gn,, we have

ni(O) = Kni(e)Vn(e)Kni(0) C X\tj (O)IK,Ii(e)12.

Thus, lemma (1) and assumption (3) imply that A>ni() converges to zero a.e. in [-rr, wr]. Lemma (2) and lemma (3) imply that

lim Kni(L)gn = 0

in mean square. With no loss of generality we can assume that

Assumption (A): \Xn(O) 2lforanyj,n,andeE[--rr,nj]. Indeed, possibly by embedding L2(Q, 9 P) into a larger space, we can

assume that L2(Q, Y; P) contains a double sequence J(it, i E %, t E Z] such that, firstly, i, is orthogonal to the u's and the i's at any lead and lag, and, secondly, 4+n={j>, t E Z}, where

+tnt =((+It +2t * *nt) %

is orthonormal white noise. Defining (it = kit + 'it, and putting

Yit += it, (11)

for i E N and t E Z, we have:

(1) Model (11) fulfills assumptions (1) through (4), with

n(0) = Yn(O) + I,, 9 ityl(o) = Y;,(0) + Ins

and therefore Xt j(=) -

+ 1, x\YI(e) = xnj(e) + 1. Moreover, Prij = Pnj for any n and j, so that Kyni = Kni for any n and i.

(2) As a consequence, if we prove proposition (2) for the y's, i.e. if

lim K,i(L)yn, = Xitg

then the desired result

lim K,,i(L)x,,t = Xit

follows, since limn --)Kni(L)+,t = 0 by lemma 3.




Under assumption (A), the function pq1(e) = [xnj(0)]-112 is defined for any 6 E [-ir, -i], is bounded and therefore has a mean-square convergent Fourier representation. Let us denote by p,q(L) the resulting square- summable filter. Now consider the vector of the first corresponding q normalized dynamic principal components:

Wnt = (Wn l,t Wn2,t ... Wnqj,t)

where W,,j,t = itn(L)pnj(L)xnt. The vector process {Wnt, t E Z} is an orthonormal q-dimensional white noise.

Lemma (4): Consider the orthogonal projection of Wnt on =

span(uj, j= 1,...,q,tE ):

(Wnl,t W,,2,t ... W,,q,t)' (12) = An(L)(ult U2t . . . Uqt)' + Rnt(

where An(L) is an n X n two-sided square-summable filter and Rnt is orthogonal to &3. Then: (A) the spectral density of Rn, converges to zero a.e. in [-ir, ir]; (B) R,t converges to zero in mean square; (C) considering the projection of ut on the space spanned by the leads and lags of Wnt,

(Ult U2t .*-- Uqt)

(13) = A,,(L-1)(W,,It Wn2,t ... Wnq,t)' + Snt(

where the spectral density of Snt converges to zero a.e. in [-nr, ir] and S,,t converges to zero in mean square.

Proof. Firstly, observe that

Wnp = pn1(L)p,1j(L)Xnt + p,q(L)Pnj(L)9nt

Because the x's belong to h?6 and the k's are orthogonal to ?<, ,puj(L) pnj(L) Er,t is the residual of the orthogonal projection of Wnp on ?0/. By assumption (3), the spectral density pnj of ,p,q(L) pnj(L)E, satisfies

fnj(o) C5 (pUnj(o))2 I p,lj(o) |12A = (Xnj(O)) -I A.

By assumption (4),fni(e) converges to zero a.e. in [-rr, -r]. Moreover, by assumption (A), fnj(e) ! A, so that the Lebesgue dominated convergence

theorem applies and fY,fnj(O) de converges to zero. Thus (A) and (B) are proved. To prove (C), from equations (12) and (13) we obtain

Iq = An(eì)An(ei") + j.(E)=A,.(eiO)An(e-i) + Ys(O),

where JR (0) and ls(O) are the spectral density matrices of Rnt and Snt, respectively. By taking the trace on both sides and noting that the trace of An(ei6)A,,(e-iO) is equal to the trace of An(e-iO)An(ei6), we get

trace (1s(O)) = trace (JR(o)).

The result follows. QED Note that lemma (4) proves that the space spanned by the normalized

dynamic principal components, which is identical to the space 2,n spanned by the dynamic principal components themselves, converges to 21, not that Wnt converges to any particular orthonormal white noise in 26. Indeed, it is easy to provide examples in which the variables Wnjl,t though converging to &2, do not converge to any vector of & . What is stated in proposition (2) is that the projection of xi, on h'In, i.e. Xit,nq converges, and that the limit is Xit

Proof of Proposition (2): We have

xit = Xit + tit = Xit,n + tit,nq (14)

with

Xit,, = K (L)x , = K (L)x,T + K,lk(L) , (15)

Combining equations (14) and (15), we obtain

[Xit - KNi(L)Xnt] + [tit -tit,j = Kni(L)gt. (16)

The spectral density of the right side of equation (16), has been denoted by Ani(0) (see the comment under lemma (3).) Because tit is orthogonal to the x's at all leads and lags,

A'ni(=) = 3,ni(0) + .i(e) -

(91 denoting the real part of a complex number), where cWni(0) is the spectral density of Xit - Kni(L)Xnts, KJ() is the spectral density of ti -

kit,nq and 9fni(0) is the cross spectrum between kit,n and Xit -Kni(L)Xn In the comment under lemma (3), we have explained why Ani(0)

converges to zero a.e. in [-7r, 7r]. If we show that ??ni(O) converges to zero a.e. in [-7r, 7r], then both _,ni(0) and Kni(e) converge to zero a.e. in [-rr, 7r], and, because both are obviously dominated by integrable functions, by the Lebesgue dominated convergence theorem, the integrals of X ni(() and Kni(e) converge to zero and the result is obtained.

Thus, we must show that the cross-spectrum between tit,n and Xit - Kni(L)Xnt converges to zero a.e. in [-nr, nr]. Consider firstly the cross- spectrum between tit,n and Xi,. Setting bi(L) = (bil(L) bi2(L) ... biq(L)), and using equation (13), we have

Xit = bi(L)(ult U2t ... Uqt)'

= bi(L)An(L-)(Wn1,t Wn2,t . . . Wnq,t)' + bi(L)Snt.

Because tit,n iS orthogonal to the terms pn1(L)xn,, for j = 1, ... . q, at any lead and lag, it is also orthogonal at any lead and lag to the terms Wnp. Thus, the cross-spectrum between .it,n and Xit is equal to the cross-spectrum between titn and bi(L)Snt , call it 9ni(O). The squared modulus of 9`ni(0) is bounded by the product of the spectral density of kit,nq which is dominated by the spectral density of xit, and the spectral density of bi(L)S,t, that is, by

bi(e - i0)Jns( )i(ei).

By lemma (4), all the entries of ls(O) tend to zero a.e. in [-Tr, Tr], so that &n(0) tends to zero a.e. in [-nr, nr].

Using the same argument, considering the cross-spectrum between tit,n and Kni(L)Xnl, we end up with the cross-spectrum between tit,n and Kni(L)B,(L)S,t, where Bn(L) is the n X q matrix having the vectors b,(L), s = 1, ... . n, on the rows. As for the spectral density of Kni(L)Bn(L)Sn,, first observe that, because lx(E) = Bn(e'i0)Bn(ei0) and En(0) = Y-X(O) +

In (0) 9

K,ni(0)B,n(e-i')Bk(e"0)k,i(0) =Knli(0)Yx(0)K,ni(0)

sKnli(OAJ)X()kni(O) q

= f Pn ,i(0) 12x\nj(e), j=1

which is bounded by the spectral density of xit. (See lemma (1).) Next, observe that the maximum eigenvalue of ls(e), which is a continuous function of the entries, tends to zero a.e. in [-Tr, Tr]. The result then follows from the inequality

Kni0)Bz e -0)nS(0)Bnl(ei1)K,ni(0)

xs AI (O)Kni(O)Bn(e-i0)fB,(ei')fKni(O)-

QED

Proof of Corollaries (1) and (2): Corollary (1) is trivial. For corollary (2), suppose that there are q factors but we project on the first q + s = q* dynamic principal components. Then

X -Xi ,,n = Pnq+1,i(L)PTq+ 1(L)Xnt + * + f5nq+s,i(L)Pnq+s(L)XntT




Because different dynamic principal components are orthogonal at any lead and lag,

Var (X *tn - Xit,n) ' __ xnq+ I(e) de + + _ X,nq+s(O) dO.

The result follows from assumption (3). QED

Proof of Proposition (3): As mentioned at the end of Section III, K 1T (L) and xnt, for fixed T (and n), are mutually dependent. Their dependence structure however is pretty intricate, and hardly can be explicitated. The random variables J4M=T)-M(T)(Knih - Kni,h)xn,t-h are not identically distributed, since boundary effects imply that the joint distributions of (KnO,2 Xnt) are not the same for t = 1 or t = T as for t = (T/2), for instance. However, such boundary effects are asymptotically negligible for central values of t, satisfying (7) for some a and b. More precisely, there exists a T3 T3(n, 8, -q) such that, for all s, t E [at, bT],

hM(T) (KT,h -K,,,h)Lhxft) 2 SUp

I -E (K T -K K,-iih)L" XnSt I sup IKTi(0) -Kni(0)s K 8s K K

ItM(T) ni

_2

[ M(T) 2

-E s ( - )Lh X. I sup fK, ,)-K ni- (0)I * s h=-M(T) OE=[-hr,Mr]

Averaging over s E- [aT, bT], it follows that, for T77? T3(n, 8, E2rl/256),

M(T) bT -

M 2 ([17) T - K Isu KnT,(0) -K 0

X(bT-aTf (Kni Ehni,hiXn,t ,h- KIis)Lpni |

s=aT Th-M(T) r,rr

2 T MT rMT) MT

+ - = EI~ [ (KTk- Kni,k)h ,kl 256 sk=-M(T) I=-M(T)

X (nTil Knil t -8] 256'

where

J 1 bT K |k and |k1X|' M(T) 2 Mkor|l| >TM(T)

Letting s' = s -=k, the matrices Knk i for k nikM(T), l M(T), and I-k ? 0 take the form

261 bT-M(T)

bT- aTEsaT+M(T)

1 aT+M(T)+k-1 1

bT-aT z ,s*,,sl+bT-aT

bT

X s Xt skX t s- = frk + k + MY, say; s=bT--M(T)+k+ 1

whenever Ik| ' M(T), 11 --- M(T) but 1 - k < 0, letting s" = s - 1, the same matrices similarly decompose into

1 bT-M(T) 1

fl1d bT - aT S +M(T) bT- aT

aT+M(T)+I-1 bT

X E X, S-kXn,s-i + : Xns_kXrl,s-l s=aT bT - aT s=bT-M(T)+I+1

= nT(k) +

iTkl + Ylk say.

The conditional expectation in the right-hand side of (17) similarly decomposes into

M(T) M(T)

E T z (Knik- K _i,k)Fnj>k)(KT,l - ... n k=-M(T) 1=-M(T)

r M(T) M(T)

+ E [ - (Kni,k - Kni,k) 01l(Kni,l -

Knij) ... < .

k=-M(T) 1=-M(T)

M(T) M(T) T

-VT2 T + E (Kni,k- Kni,k @n,k1(Kni,1 Kni,l) . . . S

k=-M(T) 1=-M(T)

= En + E, + En , say.

Due to the fact that the sums (running over s) defining Tn,kl V = 1, 2, which are divided by (bT - aT), either involve M(T) + k s 2M(T) or M(T) + 1 s 2M(T) terms, the corresponding expectations ET, v = 1, 2 are "small" as T - oo. More precisely, assuming that T ? Tl(n, ), 2)' we have (denoting by 5XaT a sum running over s, either from aT to aT + M(T) + k - 1, or from aTto aT + M(T) + 1 - 1, depending on the sign of ( -k)),

1 [M(T) M(T) n n TI E T (K, -

bT - aT k=-M(T) 1=-M(T) u=1 v=1

X (n,s-k)(t,-v(ns Knilv...sj

s=aT

282 n n aT+M(T)+k-1 M(T) M(T)

bT-T E I I E [ (n,s-k)u(Xn,s-1)v1] bTa- u=1 v=1 s=aT k=-M(T) 1=-M(T)

282 n22M(T)(2M(T) + 1)2 max var ((xn,t)u). (18)

bT- aT I1-<wn

The assumption that Tis larger than T1(n, 8, 2) has been used in substituting twice the unconditional expectations Et.. .] for the conditional ones E[... I . . --- 8]; in view of (6), T 2 T1(n, 8, -) indeed implies that P[supoE[r,r] K Ti (0) -Kni (0) | 81] S 2 so that

ni 2~~~I E[ I (X,.,-k)u(Xn,s-,)V I .. <- 8

E[ I(Xn,s-k)u(xn,s-l)v1] E[ I(Xn,s-k)u(xn,s-l)v1 > 8i]P[ > 8]

P[supOE[_,,T,TIK T(8) - K~() P [SP0E1S,TS,T]| ni (- ni (O) 8 a

E[ I (Xn,s-k)u(X,,s-l)vl]

P[supOE[-T,T]IKT.(0) - Kni(0)I s ]

-- 2E[ I(Xn,es-k)J(Xn,s-z)V1]-

It follows from (18) that E TI is 0(82M3(T)/T) as T - oo. A similar conclusion holds for ER. Hence, in view of the rate assumption on M(T),




we may conclude that there exists T4(n) and a constant M*(n) such that T > T*(n, 8) = max (T4(n), T1(n, 8, 2)) entails E + ET (n).

Turning to the main term EnT, note that F T*,0 h E = is an empirical autocovariance function, in which each covariance matrix is computed from the same number (bT - aT - 2M(T)) of observations. In contrast with T an ,kl, i based on sums which involve a number of tenns of the order of T, and are not "small" compared with bT - aT. Associated with this empirical autocovariance function is the empirical spectral density ~T* with dynamic eigenvalues X T\*(0), j = 1 . ... n. The properties of dynamic eigenvalues imply that

M(T) M(T)

~~j - ~~~ T* T -K~ E k(K K Inij 1k=-M(T) 1=-M(T)

s<E [8)2 f: A,T*1 (9) dO * ? 282E [f:1 (() do]

provided that T 2 T1(n, ., 2) (the factor 2, as in (18), is due to the substitution of unconditional expectations for the conditional ones). Now, the random sequence fT, (0) dO is a.s. bounded by

t-rr [1, (0)] dO = tr [fr [ *()] do]

n bT-M(T)

= tr [F,,,O] = (bT - aT1 I (x i= I s=aT+M(T)

(tr (F) stands for the trace of F). Hence (noting that E[(Xns )2] does not depend on s),

L.-ir J L~ a)

n bT-M(T)

r \T*(0) dO ' E i=I s=aT+M(T)

bT - aT - 2M(T) + 1

bT - aT tr (F,,,O) < tr (F,,O)

for all T. It follows that, for T T, (n, 8, ),

EnT* - 282E [f 7"TX(O) do] s 282 tr (F nO).

Summing up, for any E, -q, 8 > 0, we have shown that

P [IK (L)x,z,t- xit > E] - + - + -

16 (E2r + -I- + 82(M*(n) + 2 tr (FnO)) E225

provided that n 2 NO(E, 'q) and T > T* = TO*(n, E, , 8), where

T* = max (T, (n, 8,) T1 (n, 8), T2(n, E, 'q),

T3 n, 2^6 T4*(n, 8), Ts*(n, 8)

Proposition (3) follows, with To(n, E, -q) = T*(n, E, -q, 8), letting

E2a

82 = 82(n, E, Ti) = 256(M*(n) + 2 tr (F,,o))

QED



The Generalized Dynamic-Factor Model: Identification …pages.stern.nyu.edu/~dbackus/Identification/FHLR_factor_REStat_00.pdf · The Generalized Dynamic-Factor Model: Identification

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